The 1st method uses complex Fourier series and a regularization scheme via the Riemann-\(\zeta \) -function. The evaluation of path integrals by a real Fourier Sine Series is a frequently used method[1]} where quantum fluctuations around the classical path are expanded in terms of normalised eigenmodes \(\phi _{n}(t)\)  of the form:
\(\begin{split}\phi _{n}(t):=\sqrt{\frac{2}{T}}\sin {\frac{n\pi }{T}t}, \quad n\in \mathbb {N}\end{split}\) 
